Calculating reliable impact valve velocity by mapping instantaneous flow in a reciprocating compressor Gunther Machu, HOERBIGER Ventilwerke Ges.m.b.H & Co. Kg., A- 1110 Wien, Braunhubergasse 23
Abstract Calculated impact valve velocities can be more than twice that of measured impact valve velocities depending on the gas composition and compressor design. An inability to accurately model impact velocities can lead to dramatic consequences in valve design. State of the art methods for calculating valve impact velocities assume that cylinder pressure is homogeneous throughout the compression cycle. In this research, the pressure waves that are developed in the cylinder of a reciprocating compressor are mapped. It is demonstrated how these pressure fluctuations reduce the differential pressure across the valve and thereby reduce the valve impact velocities so low that in some cases actually prevent the valve from opening. The effect of instantaneous flow is most significant for large-bore, shortstroke compressors, high speed compressors and for heavy gas (propane) applications.
1. Introduction From time to time, interesting phenomena are observed with reciprocating compressors. In the course of e.g. a snap shot monitoring, or pV-diagram measurements unexplainable high cylinder pressures can show up, the measured suction or discharge chamber pressures won’t correspond to the indicated pressure and the re-expansion line could exhibit a wavy pattern. In very extreme applications it even has happened, that the piston rod smashes against the case with every revolution of the crankshaft. What do all these seemingly isolated observations have in common? They can be caused by wave propagation (instationary flow) inside the cylinder. Waves running back and forth across the piston produce a bending moment and thereby cause a displacement of the piston rod. Wave propagation in a heavy gas application can lead to a delayed development of the flow, inhibiting the gas exchange, thus leading to an overshoot of indicated pressure. A trapped pressure wave inside the cylinder, traveling back and forth is visible by a wavy pattern in the re-expansion line of the pV diagram. Much more important in this context is a phenomenon, which has chased valve manufacturers from the very beginning of valve dynamics simulations: the inability to correctly predict the impact velocity of the valve sealing element against the guard. The impact velocity against the guard directly translates into mechanical stresses in the sealing element material. The impact also leads to a dynamic stress increase in the coils of the valve’s closing springs. To much dynamic stress causes coil contact, promotes wear and subsequently a failure of the closing springs. The reliable prediction of the impact velocity of the sealing element against the guard is therefore the most critical issue in designing compressor valves!
1.1 State of the art The classical theory of calculating valve dynamics (and impact velocities) dates back to 1949 and was established by M. Costagliola. He assumed the cylinder pressure to be homogeneous with respect to space, and used a quasi – stationary, isentropic change of state of the gas inside the cylinder for his valve dynamics calculations. Hence, the calculation assumed an average cylinder pressure at any time t. In essence, this theory and the assumptions haven’t changed significantly throughout the years, and is still state of the art today. This method unfortunately leads to calculated impact velocities, which in many cases are more than two times higher than those from detailed field measurements. Basically, one could live with that fact, because all what is needed to obtain realistic values is to divide the calculated velocities by an average empirical factor, which is derived by a couple of measurements. This is exactly what successfully has happened in the past. In recent years it was observed though, that this empirical factor could show dramatic deviations. The spectrum of possibilities obtained from measurements leads from a factor of almost one (the calculated velocity is pretty equal to the measured one) to infinity (no impact on the guard, although the calculation reads a value)! This deviation is most significant for large-bore, short-stroke compressors, high speed compressors and for heavy gas (e.g. propane) applications. Hence, the key factor in obtaining a reliable calculation result is the ability to map the wave propagation. The industry trend to high speed compressors has therefore lead HOERBIGER to develop a new method to calculate the instationary flow situation inside the cylinder.
1.2 Existing literature E. Machu(1) identified the instationary flow inside the cylinder as the key to explain the phenomena mentioned above. In his paper he used the method of characteristics to explain abnormal high indicator pressures, and how instationary flow effects lead to additional power losses. A drawback of his approach is the inability to accurately represent the cylinder’s geometry, and furthermore no wave reflections were modeled. Thus, no wavy re-expansion patterns could be calculated. The problem of accurately predicting valve impact velocities is not addressed, but E. Machu(1) gives a hint: he explains, how the wave propagation leads to a significantly reduced differential pressure (= the driving force of valve dynamics) across the sealing element. Other publications are listed in (1), but these do not add much to the overall picture. L. Böswirth(3) suggests to include instationary flow effects during the opening of a valve by using a modified Bernoulli equation to describe the gas exchange. In fact, that improves accuracy, but not really satisfying: the above mentioned factor of calculated to measured impact velocity will decrease from e.g. 2.5 to 1.9 – obviously, the physical situation is not correctly represented by this approach. In this paper it will be shown, that the instationary flow model developed by HOERBIGER is able to cover all of the above mentioned phenomena with a level of accuracy, which leaves no open questions.
2. The simulation model Figure 1 exhibits the model used in the calculations. There are three interconnected calculation domains, the suction and discharge chambers, and the cylinder. One could immediately argue, that the exact geometries are not available in daily business, and therefore restrict the calculation to the very few cases, where drawings are available. This is not the case, as will be shown in the following. With a few assumptions and standard data like bore, stroke and valve pocket diameter very good simulation results are obtained.
Fig. 1: The simulation model
For the valve cage a height of 1.5 times the valve pocket diameter is used, the passage area to the cylinder is assumed to be 0.5 to 0.3 times the valve area, and the cylinder geometry is defined by bore and stroke. By this information, the geometry is complete. In the course of the calculation, the piston will increasingly diminish the valve passage area. This effect is included in the simulations.
2.1 Governing system of equations The state of the gas in the cylinder is determined by two factors: on the one hand from the kinematics of the drive, which determines the volume inside the cylinder (= area(x,t) at each node point) and therefore the isentropic change of state of the gas. On the other hand from the in- and outflow into the nodes from neighboring nodes, or the flow through the suction or discharge valve at the boundary. The piston position z(φ) with respect to crank angle is given by
z (ϕ ) =
Vs r + r (1 − cos ϕ ) + (1 − 1 − λ 2 sin 2 ϕ ) AP λ
(1)
where r denotes the crank radius, λ the rod ratio, Vs the clearance volume, and AP denotes the piston area. In order to calculate the velocity u(x,t) and the pressure p(x,t) at every node point at time t, on has to solve the instationary Euler equations of fluid dynamics (one dimensional, frictionless, in conservative form, please review J.D.Anderson(2)): Equation of continuity:
∂ ( ρA) ∂ ( ρAu ) + =0 ∂t ∂x
and
Eulerequation:
∂ ( ρAu ) ∂ ( ρAu 2 + pA) ∂A + =p ∂t ∂x ∂x
(2)
In (2) ρ(x,t) denotes the density with respect to space and time, u(x,t) and p(x,t) the velocity and pressure, and A(x,t) is the area at the position x. As the change of state of the gas is assumed to be isentropic, no additional energy equation has to be solved. In order to solve (2), there a lots of first and second order methods in the literature. Here, a variant of the Lax – Wendroff method, namely the method of MacCormack is used (2. order with respect to space and time). It is assumed, that all variables are known at time t, and the flux terms Uj (in brackets in (2)) have to be calculated at the position x for the time t+∆t. This is done as follows:
⎛ ∂U j ⎞ ⎟ ∆t (5) U tj+ ∆t = U tj + ⎜⎜ ⎟ ⎝ ∂t ⎠ av where (..)av is a representative average of the time derivative, calculated by a predictor – corrector schema. After performing the integration, the variables ρ(x,t), u(x,t) and the pressure p(x,t) are decoded from Uj. With the calculated pressure, the differential equation of the valve sealing element is solved:
&= η∆pFO − F m SE Y& Springs
(6),
where Y denotes the lift, η is the lift dependent drag coefficient, ∆p the differential pressure across the sealing element and FO is the pressurized area. FSprings denotes the closing spring force, and mSE the mass of the sealing element.
2.2 Boundary conditions If a valve is in the open position, the velocity in the gap due to the pressure ratio across the sealing element is given by:
⎛p ⎞ 2κ p 1 − ⎜⎜ K ⎟⎟ u= (κ − 1) ρ ⎝ p ⎠
κ −1 κ
,
(7)
the corresponding mass flow [kg/s] is calculated by:
m&= Aeff ρ K u
(8)
Aeff denotes the effective flow area of the valve at time t in [m2], and is a function of the valve lift. The mass flow from (8) is used as the boundary condition for the flow dynamics calculation (meshes (1) and (2), or (1) and (3) in figure 1), and is positive or negative depending on the pressure ratio. The pressure is extrapolated from the neighboring nodes. If the valves are closed, then the mass flow is zero, and the pressure is extrapolated from the neighboring node points again.
2.3 Spatial and temporal discretization An analysis of stability of the governing equations (2) reveals a stability criterion for the spatial and temporal discretization, the Courant Friedrichs Levy criterion (CFL):
∆t = C
∆x , max( a + u )
(9)
where a is the local speed of sound, u is the local fluid velocity and C is the Courant number. For a Courant number C ≤ 1 stable solutions of (2) are obtained. In the computer code, equation (9) is evaluated at every node point on the x – axis to determine the lowest value of ∆t, which is then applied to the whole field at t+∆t.. A spatial discretization of 0,5[mm] was used, hence the time step follows from (9). From these values follows a calculation time for a full revolution and a light gas of around 30[s] on a 800MHz laptop computer.
3. Simulation vs. measurements 3.1 Waves in the re-expansion line Especially for high speed compressors with a high bore / stroke ratio (e.g. 2.4 for the compressor in figure 2) compressing a heavy gas, often a very unpleasant fact becomes obvious: during the discharge event, the indicated pressure significantly exceeds the line pressure (e.g. 40% in figure 2) for no obvious reason. In such a case, immediately the valves are made responsible for not working properly, being not enough efficient or not being properly designed. It is not the valve’s fault! Simulating the instationary flow inside the cylinder immediately reveals the explanation. In figures 2, 3 and 4, a propane compressor is depicted which was the focus of HOERBIGER measurements back in 1992 (Union Pacific Resources, TX, USA). At this compressor, the valve dynamics as well as the indicated pressures at two opposite sides of the cylinder were simultaneously recorded. This was achieved by drilling holes through the center screws of both a suction and a discharge valve, and inserting pressure transducers planar with the cylinder wall (at the second stage, 346.1 [mm] bore, 139.7[mm] stroke, 1190[rpm], 3.9 – 12.6[bara). The result is exhibited in figure 3: the indicated pressure during the discharge event is drastically different at the two opposite cylinder sides (suction side (1), discharge side (2)) for the same time t! At the suction side, the information that the discharge valve has opened (see figure 3, the first pressure decrease in line (2)) is delayed. In fact for exactly the amount of time it takes the pressure wave to travel across the piston with the speed of sound. Hence, the cylinder pressure at the suction side (1) will increase until the expansion wave has reached the suction side. This leads to the observed overshoot in pressure. The expansion wave starts to accelerate the flow, hence velocity is build up and then the cylinder pressure is relieved by the flow through the discharge valve. Now, if indicator measurements would be done at this location without knowing about the involved physics, they would wrongly indicate very high valve losses! Here, also another fact becomes clear: the pressure gradient across the piston (more than 3.6 [bar] in figure 2) produces a significant bending moment on the piston rod. This explains,
why in extreme applications it has happened, that the rod touches the casing with every revolution (and subsequently is prone to fail). Very nicely also the wavy pattern in the re-expansion line is visible, here the simulation very closely matches the measurements (figure 3). The frequency of the ‘trapped’ pulsations in the cylinder corresponds exactly to the time it takes a pressure wave to travel twice across the piston (back and forth). E. Machu(1) hints at how to avoid these pulsations: a tapered piston (producing more room close to the discharge side) reduces the involved flow velocities (and therefore the pressure amplitudes) with only a slight increase of clearance volume. 19 17
1
2
[bar] Pressure
15 13 11 9 7 5 3 1 180
270
360
[°] Crank Angle
Fig. 2: Union Pacific, second stage, simulated indicator pressures at two opposite side of the cylinder: (1) in the center screw of the suction valve, (2) in the center screw of the discharge valve. If indicator measurements would be done only at location (1), it would (wrongly) seem that the valves are responsible for the high overshoot in pressure!
1
2
Fig. 3: Union Pacific, second stage, comparison with measurements from 1992: the two measured traces (copied) on the left and right hand side show the cylinder pressure at two locations, at the suction side (1) uppermost line, and on the discharge side (lower trace). Superimposed are the simulated curves from figure 2.
3.2 Valvedynamics The inertia of the flow leads to another phenomena: the discharge valve (trace 3 and 4 in figure 4) starts to close slightly before the top dead center (TDC, 360° crank angle) and would almost perfectly be closed at TDC, but the outward gas flow which has developed can’t be stopped so easily. Thus, the valve is forced to open once more (the bump after TDC in figure 4), even though the piston has already reversed its direction! Figure 4 also shows the measured valve sealing element motion (3) superimposed to the simulated curve (4) – a rather good agreement. The overshoot of the measured valve motion (3) is an artefact of the measuring method.
3
4
Abb. 4: Union Pacific, second stage, measured (3) and simulated (4) discharge valve motion. Due to the inertia of the heavy gas, even after top dead center (TDC, and reversed piston motion) there is still an outward flow, which inhibits a proper closing of the valve around TDC.
The inclusion of instationary flow effects also solves the mystery of calculating reliable impact velocities of the sealing element against the guard. As exhibited in figure 2, the assumption of a homogenious cylinder pressure (= everywhere the same pressure) in the classical theory fails. Due to wave propagation effects, the differential pressure across the valve sealing element is “eaten up” at the instant of opening to overcome the gas inertia. Compared to the differential of average cylinder pressure to discharge chamber pressure, the instantaneous differential across the sealing element is drastically reduced, leading to a much lower initial acceleration of the sealing element (see equation (6)). As the free flight phase of the sealing element (the time it takes the sealing element to travel from seat to guard) is very short (typically less than 0.5[ms]), a second factor comes into play: will the expansion wave, initiated by the opening event of the valve, manage to reach the valve again in its free flight phase (after reflection from the closed cylinder end)? If not, then the valve dynamics are mainly driven by the initial reduced pressure differential, and the impact velocity against the guard will be rather low (typically the case for heavy gases, with low speed of sound and large bore cylinders). If yes (e.g. for light gases and small bore cylinders), the resulting pressure differential across the sealing element is abruptly increased, because the initial expansion wave has converted into a compression wave due to the reflection on the closed cylinder side. Correspondingly, the impact velocities will be rather high.
Examples for both ends of the spectrum are given in figure 5:
(a) (b) (c) Fig. 5: (a) Simulation of valve dynamics for a hydrogen application, and (b) measurements for the same application. (c) Simulation for a propane compressor: here, the inertia of the flow uses up so much pressure differential, that actually the valve is prevented from opening properly. Measurements (not shown) confirm the findings.
In figure 5 (a) and (b), valve dynamics simulations and measurements are shown for a hydrogen application. Also here, the instationary flow calculation compares very good to measurements, and predicts almost exactly the measured impact velocity (within measuring tolerance). In line with the arguments given above is the fact, that there is almost no difference between classical theory and instationary flow calculation. The velocity of sound is so high, that the cylinder pressure is almost homogeneous. Figure 5 (c) exhibits once more a calculation for a propane application. Here, the valve only opens halfway (~50% lift). Measurements from the past confirm that result. Classical theory would calculate an impact velocity of 5 [m/s], resulting in a factor of calculated / measured velocities of infinite (see section 1.1)!
4. Concluding remarks Instationary flow effects in the cylinder can lead to rather severe problems, on the compressor’s side (power losses, piston rods hitting the casing) as well as on the valve side (late closure, reliable prediction of impact velocities - valve design). Especially the trend towards higher compressor speeds in combination with a large bore / stroke ratio requires the inclusion of instationary flow effects in the design phase of the compressor and the valves. The simulation technique developed at HOERBIGER requires no additional information than that already known from daily business, and the runs within seconds on an ordinary desktop pc. It definitely would make sense to use this new technique together with compressor manufacturers already in the design stage of a new compressor as it could eliminate unfortunate surprises from the very beginning.
References (1) E. Machu, Problems with modern high speed short stroke reciprocating compressors: Increased power requirement due to pocket losses, piston masking and gas inertia, eccentric gas load on the piston. Gas machinery conference Denver / USA, Oct. 1998 (2) J.D. Anderson , Computational Fluid Dynamics, 1995, McGraw Hill (3) L. Böswirth, Strömung und Ventilplattenbewegung in Kolbenverdichterventilen, Erweiterter Nachdruck 1998, Eigenverlag Wien
